: In certain areas of artificial intelligence there is need to represent continuous change and to make statements that are interpreted with respect to time intervals rather than time points. To this end we develop a modal temporal logic based on time intervals, a logic which can be viewed as a generalization of pointbased modal temporal logic. We discuss related logics, give an intuitive presentation of the new logic, and define its formal syntax and semantics. We make no assumption about the underlying nature of time, allowing it to be discrete (such as the natural numbers) or continuous (such as the rationals or the reals), linear or branching, complete (such as the reals) or not (such as the rationals). We show, however, that there are formulas in the logic that allow us to distinguish all these situations. We also give a translation of our logic into first-order logic, which allows us to apply some results on first-order logic to our modal one. Finally, we consider the difficulty o...

We define formal notions of temporal domain and temporal database, and use them to survey a wide spectrum of temporal query languages. We distinguish between an abstract temporal database and its concrete representations, and accordingly between abstract and concrete temporal query languages. We also address the issue of incomplete temporal information. 1 Introduction A temporal database is a repository of temporal information. A temporal query language is any query language for temporal databases. In this paper we propose a formal notion of temporal database and use this notion in surveying a wide spectrum of temporal query languages. The need to store temporal information arises in many computer applications. Consider, for example, records of various kinds: financial [37], personnel, medical [98], or judicial. Also, monitoring data, e.g., in telecommunications network management [4] or process control, has often a temporal dimension. There has been a lot of research in temporal dat...

A class of interval-based temporal languages for uniformly representing and reasoning about actions and plans is presented. Actions are represented by describing what is true while the action itself is occurring, and plans are constructed by temporally relating actions and world states. The temporal languages are members of the family of Description Logics, which are characterized by high expressivity combined with good computational properties. The subsumption problem for a class of temporal Description Logics is investigated and sound and complete decision procedures are given. The basic language TL-F is considered #rst: it is the composition of a temporal logic TL # able to express interval temporal networks # together with the non-temporal logic F # a Feature Description Logic. It is proven that subsumption in this language is an NP-complete problem. Then it is shown how to reason with the more expressive languages TLU-FU and TL-ALCF . The former adds disjunction both at...

In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specic times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics. In this paper we analyze the complexity of the satisability problem of a number of hybrid temporal logics: the basic hybrid language over transitive frames; nominal tense logic over transitive frames, strict total orders, and transitive trees; nominal Until logic; and referential interval logic. We discuss the eects of including nominals, the @ operator, the somewhere modality E, and the dierence operator D. Adding nominals to tense logic leads for several frame{classes to an increase in complexity of the satisability pro...

Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the So a school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev), the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the rst technical, the second conceptual. First, we showthathybridization gives rise to well-behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full rst-order expressive strength, is demonstrated for a weaker local language capable of de ning the Until operator � we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. 1

this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science fiction writers, and were of vital concern to our everyday life and commonsense reasoning. So whatever approach to AI one takes [ Russell and Norvig, 1995 ] , temporal and spatial representation and reasoning will always be among its most important ingredients (cf. [ Hayes, 1985 ] ). Knowledge representation (KR) has been quite successful in dealing separately with both time and space. The spectrum of formalisms in use ranges from relatively simple temporal and spatial databases, in which data are indexed by temporal and/or spatial parameters (see e.g. [ Srefik, 1995; Worboys, 1995 ] ), to much more sophisticated numerical methods developed in computational geom

In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatiotemporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u , which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with muli-dimensional formalisms and discuss possible directions for further research.

This paper is a continuation of the work started in [FG92] on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component one-dimensional temporal logics to the resulting two-dimensional temporal logic. Three basic logical properties are analysed, namely soundness, completeness and decidability. Each combination method is composed of three submethods that combine the languages, the inference systems and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages with varying expressivity and varying degree of transference of logical properties. The temporalisation method and the independent combination method are shown to transfer all three basic logical properties. The method of full interlacing of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several...

We study two-dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using non-structural Gabbay-type inference rules.

The paper uses left and right neighbourhoods as primitive interval modalities to define other unary and binary modalities of intervals in a first order logic with interval length. A complete first order logic for the neighbourhood modalities is presented. The paper demonstrates how the logic can support formal specification and verification of liveness and fairness, and also of various notions of real analysis. 1 Introduction Interval temporal logics, based on ITL [11], have shown to be useful for the specification and verification of safety properties of real-time systems. In these logics one can succinctly express properties like: "for all intervals of a given size, OE must hold", and "if OE holds for an interval, then there is a subinterval where / holds", and so on. However, these logics cannot express more abstract liveness properties like "eventually there is an interval where OE holds" and "OE will hold infinitely often in the future". The reason for this limitation is that the...